I don't think the interpretation of HolgerFiedler is correct, or I really don't understand why. I feel there is a confusion between "motion", angular momentum and magnetic moment...
I am more satisfied with this interpretation I have found on
https://www.physicsforums.com/threads/question-about-a-magnetic-dipole-in-an-inhomogeneous-magnetic-field-please-help.109669/
Your question has to be answered twice. First, classically (which doesn't really apply, but that wasn't known until after the experiment): The inhomogeneity can be made large enough so that there is significant deflection before much rotation takes place. You can see what is needed from the equations. You would have to assume a classical moment of inertia of the neutral particle.
Since QM is needed, this classical explanation is really irrelevant. The QM explanation: The particle (if spin 1/2) can only be either up or down in the direction of the B field. The energy difference between the two states is \Delta E=2B\cdot\mu. The spin can only be flipped by an oscillating magnetic field of frequency \hbar\omega=\Delta E. The field in the SG experiment are static, so no spin flip takes place
Classically, I think that the precession can have a really fast response, while for the rotation to occur, some momentum has to be evacuated, through damping (in the case of the electron, radiation maybe). In the case of magnets, I am not really sure how and where the angular momentum goes so fast, but I have never observed any macroscopic precession...
Edit: In the case of the big magnet, all the moments probably try to precess but are blocked because they prefer to stay aligned (shape anisotropy), it's like infinite damping, and the magnet just experiences a torque leading to its alignment along the field. Maybe the magnetic moment is converted into angular momentum via Einstein–de Haas effect, but numerically the angular momentum is too weak to be observed?